Galois orbit of Artin representations 4.2e12_3e2_5e2.8t22.16

• Label: 4.2e12_3e2_5e2.8t22.16
• Dimension: 4
• Group: $C_2^3 : D_4 $
• Conductor: $ 2^{12} \cdot 3^{2} \cdot 5^{2}$
• Frobenius-Schur indicator: 1

Basic invariants

Dimension:	$4$
Group:	$C_2^3 : D_4 $
Conductor:	$921600= 2^{12} \cdot 3^{2} \cdot 5^{2} $
Artin number field:	Splitting field of $f= x^{8} - 8 x^{6} + 33 x^{4} - 50 x^{2} + 25 $ over $\Q$
Size of Galois orbit:	1
Smallest containing permutation representation:	$C_2^3 : D_4 $
Parity:	Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 769 }$ to precision 5.

Roots:

$r_{ 1 }$	$=$	$ 26 + 666\cdot 769 + 247\cdot 769^{2} + 606\cdot 769^{3} + 263\cdot 769^{4} +O\left(769^{ 5 }\right)$
$r_{ 2 }$	$=$	$ 48 + 309\cdot 769 + 367\cdot 769^{2} + 228\cdot 769^{3} + 462\cdot 769^{4} +O\left(769^{ 5 }\right)$
$r_{ 3 }$	$=$	$ 50 + 54\cdot 769 + 236\cdot 769^{2} + 425\cdot 769^{3} + 255\cdot 769^{4} +O\left(769^{ 5 }\right)$
$r_{ 4 }$	$=$	$ 284 + 332\cdot 769 + 64\cdot 769^{2} + 674\cdot 769^{3} + 200\cdot 769^{4} +O\left(769^{ 5 }\right)$
$r_{ 5 }$	$=$	$ 485 + 436\cdot 769 + 704\cdot 769^{2} + 94\cdot 769^{3} + 568\cdot 769^{4} +O\left(769^{ 5 }\right)$
$r_{ 6 }$	$=$	$ 719 + 714\cdot 769 + 532\cdot 769^{2} + 343\cdot 769^{3} + 513\cdot 769^{4} +O\left(769^{ 5 }\right)$
$r_{ 7 }$	$=$	$ 721 + 459\cdot 769 + 401\cdot 769^{2} + 540\cdot 769^{3} + 306\cdot 769^{4} +O\left(769^{ 5 }\right)$
$r_{ 8 }$	$=$	$ 743 + 102\cdot 769 + 521\cdot 769^{2} + 162\cdot 769^{3} + 505\cdot 769^{4} +O\left(769^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation

$(1,4)(2,3)(5,8)(6,7)$

$(3,6)(4,5)$

$(2,7)(4,5)$

$(1,5)(2,3)(4,8)(6,7)$

$(1,8)(4,5)$

$(1,3)(2,4)(5,7)(6,8)$

$(1,3)(2,5)(4,7)(6,8)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 8 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,8)(2,7)(3,6)(4,5)$	$-4$
$2$	$2$	$(1,4)(2,3)(5,8)(6,7)$	$0$
$2$	$2$	$(3,6)(4,5)$	$0$
$2$	$2$	$(1,3)(2,4)(5,7)(6,8)$	$0$
$2$	$2$	$(1,2)(3,4)(5,6)(7,8)$	$0$
$2$	$2$	$(1,8)(4,5)$	$0$
$2$	$2$	$(1,5)(2,3)(4,8)(6,7)$	$0$
$2$	$2$	$(1,8)(3,6)$	$0$
$2$	$2$	$(1,2)(3,5)(4,6)(7,8)$	$0$
$2$	$2$	$(1,6)(2,4)(3,8)(5,7)$	$0$
$2$	$4$	$(1,5,8,4)(2,3,7,6)$	$0$
$2$	$4$	$(1,7,8,2)(3,5,6,4)$	$0$
$2$	$4$	$(1,3,8,6)(2,4,7,5)$	$0$
$2$	$4$	$(1,6,8,3)(2,4,7,5)$	$0$
$2$	$4$	$(1,4,8,5)(2,3,7,6)$	$0$
$2$	$4$	$(1,7,8,2)(3,4,6,5)$	$0$

The blue line marks the conjugacy class containing complex conjugation.